THE JOURNAL OF CHEMICAL PHYSICS 133, 034122 共2010兲
A homogeneous nonequilibrium molecular dynamics method for calculating the heat transport coefficient of mixtures and alloys Kranthi K. Mandadapu,1 Reese E. Jones,2,a兲 and Panayiotis Papadopoulos2 1
Department of Mechanical Engineering, University of California, Berkeley, California 94720-1740, USA Sandia National Laboratories, Livermore, California 94551-0969, USA
2
共Received 7 April 2010; accepted 10 June 2010; published online 21 July 2010兲 This work generalizes Evans’ homogeneous nonequilibrium method for estimating heat transport coefficient to multispecies molecular systems described by general multibody potentials. The proposed method, in addition to being compatible with periodic boundary conditions, is shown to satisfy all the requirements of Evans’ original method, namely, adiabatic incompressibility of phase space, equivalence of the dissipative and heat fluxes, and momentum preservation. The difference between the new equations of motion, suitable for mixtures and alloys, and those of Evans’ original work are quantified by means of simulations for fluid Ar–Kr and solid GaN test systems. © 2010 American Institute of Physics. 关doi:10.1063/1.3459126兴 I. INTRODUCTION
The heat transport coefficient relating heat flux to the temperature gradient in the linear irreversible thermodynamic regime is an important phenomenological parameter for the modeling of heat conduction. For single-species systems and alloys without diffusion, this transport coefficient reduces to the usual thermal conductivity tensor . Molecular dynamics 共MD兲 simulations can be employed to estimate this coefficient given the atomic structure and an interatomic potential that accurately describes the interactions among the atoms.1–7 The two most commonly used methods for estimating the heat transport coefficient are 共i兲 the direct method3,6 and 共ii兲 the Green–Kubo 共GK兲 method.2,3 An alternative to these methods, the so-called homogeneous nonequilibrium molecular dynamics 共HNEMD兲 method4,7–12 has advantages over the GK method in that it is free of difficulties involving the calculation and integration of the heat flux autocorrelation tensor, and the direct method which has to contend with strong size effects and requires unrealistically large temperature gradients. In addition, the HNEMD method typically yields better statistical averages than both the direct and the GK method at a lower overall computational cost. In the HNEMD method, thermal transport is modeled by means of a mechanical analog, where a fictitious external force mimics the temperature gradient. Using the linear response theory,5,7 the long-time ensemble average of the heat flux vector can be shown to be proportional to the external force field 共when the latter is sufficiently small兲, with the constant of proportionality being the heat transport coefficient tensor. The HNEMD method was originally developed for systems modeled by Lennard-Jones pair potentials.4 It was subsequently applied to molecular liquids with distance and angle constraints using an intramolecular four-body potential,10,11,13 and further extended to systems modeled by three-body potentials7 and recently to M-body potentials.12 a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-9606/2010/133共3兲/034122/11/$30.00
The HNEMD equations of motion were developed to satisfy three basic conditions: adiabatic incompressibility of phase space 共AI⌫兲, preservation of zero total momentum, and equivalence of dissipative and heat fluxes. These equations are suitable for single-species systems since they yield the heat transport coefficient without having to calculate any average quantities from equilibrium simulations. If these HNEMD equations of motion are employed for multispecies systems, e.g., a semiconductor gallium-nitride system,14,15 the long-time average of the heat flux vector still remains proportional to the external field. However, the constant of proportionality is not just equal to the heat transport coefficient, but involves additional correlation integrals since the HNEMD equations do not satisfy the AI⌫ condition and the dissipative flux is not equivalent to the heat flux vector as in the case of single-species systems. Hence, these correlation integrals need to be evaluated from an equilibrium simulation in addition to running the HNEMD algorithm. An earlier attempt to obtain nonequilibrium equations of motion specifically tailored to multispecies systems also requires equilibrium MD simulations in addition to running the nonequilibrium algorithm as the AI⌫ condition is not satisfied.16 As a result, the development of a nonequilibrium method to obtain the heat transport coefficient for multispecies systems without resorting to additional equilibrium simulations remains an open problem. This paper proposes precisely such a method, termed the mixture-HNEMD 共M-HNEMD兲 method. This method preserves all three conditions cited earlier and, in addition, remains compatible with periodic boundary conditions. This paper is organized as follows: In Sec. II, a review of basic irreversible thermodynamics is given within the context of heat conduction for binary mixtures and alloys. The linear response theory is derived in Sec. III for multispecies systems modeled by multibody potentials. In Sec. IV, the estimate of the heat transport coefficient for multispecies systems is analyzed for the HNEMD method. Next, the M-HNEMD equations of motion are developed in Sec. V,
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followed by an application of HNEMD and M-HNEMD equations of motion to a pure argon 共Ar兲 system, an argonkrypton 共Ar–Kr兲 binary mixture, and a perfect galliumnitride 共GaN兲 crystal in Sec. VI.
II. LINEAR IRREVERSIBLE THERMODYNAMICS
Following the theory of irreversible processes presented in Ref. 17 共Sec. 44兲, the total rate of entropy production for a mixture of n species capable of diffusion, heat conduction, and the cross phenomena 共Sorét and Dufour effects兲 in the absence of external forces on any individual species is given by n
= 兺 Ji · Xi + JQ · XQ .
共1兲
i=1
Here, Xi = −共i / T兲 and XQ = −T / T2 are the generalized forces driving the irreversible processes of diffusion and heat transfer, respectively, where i is the concentration of species i and T is the thermodynamic temperature. Also, JQ is the heat flux vector and Ji are the diffusive flux vectors, which satisfy n
Ji = 0. 兺 i=1
共2兲
Taking into account Eq. 共2兲, the total rate of entropy production can be equivalently expressed as
LQQ =
V kB
冕
⬁
˜ 共t兲 丢 ˜J 共0兲典 dt, 具J Q Q c
共9兲
0
where V is the volume, kB is the Boltzmann constant, and 具 · 典c denotes the average over equilibrium phase space distribution. It is important to emphasize that in the case of binary liquid mixtures the evaluation of LQQ is not sufficient for estimating the thermal conductivity . Indeed, experiments measuring thermal conductivity of binary liquid mixtures are typically conducted using the stationary state 共J1 = 0兲. In this −1 L1QXQ, which reduces case, Eq. 共5兲 implies X1 − X2 = −L11 Eq. 共6兲 to −1 JQ = 共LQQ − LQ1L11 L1Q兲XQ .
共10兲
Therefore, for binary liquid mixtures, admitting Fourier’s law in the form JQ = − T and recalling the definition of XQ leads to
=
1 −1 共LQQ − LQ1L11 L1Q兲. T2
共11兲
In contrast to binary liquids, if in a perfect crystalline solid no significant mass diffusion exists 共e.g., GaN at room temperature兲 one may neglect diffusion and the cross phenomena. In this case, XQ is the only significant generalized force, therefore the heat flux vector in the linear irreversible regime follows the phenomenological law JQ = LQQXQ, thus yielding the thermal conductivity as = 共1 / T2兲LQQ.
n−1
= 兺 Ji · 共Xi − Xn兲 + JQ · XQ .
共3兲
i=1
Reducing Eq. 共3兲 to a binary mixture 共n = 2兲 results in
= J1 · 共X1 − X2兲 + JQ · XQ .
共4兲
Assuming that the irreversible processes are close to thermodynamic equilibrium, the linear phenomenological relations J1 = L11共X1 − X2兲 + L1QXQ ,
共5兲
JQ = LQ1共X1 − X2兲 + LQQXQ ,
共6兲
are proposed, where L11, L1Q, LQ1, and LQQ are phenomenological coefficient matrices that satisfy Onsager’s reciprocal T T T = L11, L1Q = LQ1, and LQQ = LQQ, see Refs. relations L11 17–19. Herein, superscript T denotes the matrix transpose. Setting J1 = ˜J1共t兲 and JQ = ˜JQ共t兲, where ˜J1共t兲 and ˜JQ共t兲 represent the phase functions J1 and JQ as functions of time 共see Sec. III兲, and following the derivations in Ref. 20, the Green–Kubo relations for L11, L1Q, and LQQ are given in terms of the respective correlation functions by L11 =
L1Q =
V kB
冕
V kB
⬁
˜ 共t兲 丢 ˜J 共0兲典 dt, 具J 1 1 c
共7兲
˜ 共t兲 丢 ˜J 共0兲典 dt, 具J Q c 1
共8兲
0
冕
⬁
0
III. LINEAR RESPONSE THEORY
In linear response theory 共Ref. 5, Chap. 5兲, the general form of the equations of motion, when perturbed by an external field Fe in the presence of a Nosé–Hoover 共NH兲 thermostat, is r˙ i =
pi + Ci共⌫兲Fe , mi
p˙ i = Fi + Di共⌫兲Fe − pi , 1 ˙ = Q
冉兺 N
i=1
共12兲
冊
pi · pi − 3NkBT , mi
where mi is the mass of atom i and ⌫ = 兵共ri , pi兲 , i = 1 , 2 , . . . , N其 is the phase space with ri and pi being the position and momentum vectors of the ith atom in an N-atom system, respectively. Here, Fi is the interatomic force, and Ci共⌫兲 and Di共⌫兲 are the second-order tensor phase variables which describe the coupling of the system to the applied external field Fe 共Ref. 21 and Ref. 5, Chap. 5兲. Also, in the rate equation for the thermodynamic friction coefficient associated with the Nosé–Hoover thermostat,22 Q is a parameter chosen to yield the canonical phase space distribution in the absence of external force fields.23 When Fe = 0, the phase space distribution f共⌫ , , t兲 becomes the 共extended兲 canonical distribution f c,
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e−共H0共⌫兲+共1/2兲Q 兲 , f c共⌫, 兲 = Z共兲
共13兲
i = E iI +
2
where  = 1 / kBT, Z共兲 = 兰e−共H0共⌫兲+共1/2兲Q 兲d⌫d, and H0 is the total internal energy N
H0 = 兺 i=1
pi · pi + ⌽共r兲, 2mi
共14兲
see Ref. 22. The potential ⌽共r兲 that describes the M-body interactions among the atoms is of the general form ⌽共r兲 =
1 1 u2共ri1,ri2兲 + 兺 兺 u3共ri1,ri2,ri3兲 2! 共i1,i2兲 3! 共i1,i2,i3兲 1 兺 uM共ri1,ri2, . . . ,riM兲, M! 共i1,i2,. . .,iM 兲
1 1 Fii2 丢 rii2 + 兺 兺 Fii i 丢 共rii2 + rii3兲 2! 共i2兲i 3! 共i2,i3兲i 2 3
+ ¯+
1 兺 Fii ,. . .,i 丢 共rii2 + ¯ + riiM兲, M! 共i2, . . . ,iM 兲i 2 M 共19兲
also see Ref. 26. The second-order tensor i is clearly a combination of the 3 ⫻ 3 identity tensor I scaled by the energy associated with atom i and the virial associated with atom i. Assuming that the external field Fe is small and timeindependent, the distribution function f共⌫ , , t兲 can be obtained as f共⌫, ,t兲 = f c共⌫, 兲 −
冕
t
exp共− iL0共t − s兲兲i⌬L共s兲f c共⌫, 兲ds,
0
共15兲
共20兲
where M ⱕ N, uK共ri1 , ri2 , . . . , riK兲 characterizes all K-body interactions,24 and 兺共i1,i2,. . .,ir兲 denotes summation over all N ! / 共N − r兲! permutations of 共1 , 2 , . . . , r兲. Taking into consideration Eqs. 共14兲 and 共15兲, the internal energy Ei of atom i may be defined as
where i⌬L共s兲f c共⌫ , 兲 = iL共s兲f c共⌫ , 兲 − iL0 f c共⌫ , 兲, with iL and iL0 being the Liouvilleans corresponding to field-dependent 共Fe ⫽ 0兲 and field-independent 共Fe = 0兲 equations of motion, respectively 共Ref. 5, Sec. 5.1兲. Since f c共⌫ , 兲 is the steadystate solution of the equation 共 / t兲f = −iL0 f, it follows that iL0 f c共⌫ , 兲 = 0.22 Hence,
+ ¯+
Ei =
i⌬L共s兲f c共⌫, 兲 = iL共s兲f c共⌫, 兲
1 pi · pi 1 + 兺 u2共ri,ri2兲 + 兺 u3共ri,ri2,ri3兲 2! 共i2兲i 3! 共i2,i3兲i 2mi 1 + ¯+ 兺 uM共ri,ri2, . . . ,riM兲, M! 共i2, . . . ,iM 兲i
= 共16兲 =
1 ⌽ = 兺 Fii + 兺 Fii i ri 共i2兲i 2 2! 共i2,i3兲i 2 3
+ ¯+
1 兺 Fii ,. . .,i , 共M − 1兲! 共i2, . . . ,iM 兲i 2 M
共17兲
N
where
冋兺 冋兺 i=1
N
册
· C iF e + 兺 · DiFe f c共⌫, 兲 ri i=1 pi N
+ −
pi DTi m
i=1
册
N
+ 兺 CTi Fi · Fe f c共⌫, 兲 i=1
= V关B共⌫兲 − J共⌫兲兴 · Fe f c共⌫, 兲,
共21兲
where
where Fii2,. . .,iK = −共 / ri兲uK共ri , ri2 , . . . , riK兲 is the K-body force contribution on atom i and 2 ⱕ K ⱕ M. Following the Irving–Kirkwood procedure for field-free equations of motion and zero total momentum,25 the macroscopic instantaneous heat flux vector JQ共⌫兲 = ˜JQ共t兲 in the absence of external forces on any individual species is obtained in tensorial form as 1 pi JQ共⌫兲 = 兺 Ti , V i=1 mi
册
˙ ˙ ·⌫+⌫· + ˙ + ˙ f c共⌫, 兲 ⌫ ⌫ N
where 兺共i2 , i3 , . . . , ir兲i denotes summation over all 共N − 1兲 ! / 共N − r兲! permutations of 共1 , 2 , . . . , r兲 excluding i, see Ref. 12. Here, it can be seen that the sum of Ei over i yields the total internal energy H0 given by Eq. 共14兲. The total interatomic force Fi on atom i is then given by Fi = −
冋
共18兲
1 B共⌫兲 = V
冉兺 N
i=1
N
· CTi + 兺 · DTi ri i=1 pi
冊
共22兲
is the phase space compression factor 共Ref. 5, Sec. 3.3兲, and
J共⌫兲 =
1 V
冉
N
兺 DTi i=1
N
pi − 兺 C TF i mi i=1 i
冊
共23兲
is defined as the “dissipative” flux in Ref. 5 共Sec. 5.1兲. In earlier works,4,7,12,21 it was assumed that the equations of motion 共12兲 satisfy the condition of adiabatic incompressibility of phase space 共AI⌫兲, i.e., B = 0. However, in this paper, B ⫽ 0 is allowed in the derivation of the linear response theory. Using Eqs. 共20兲 and 共21兲, the long-time ensemble average of the heat flux vector ˜JQ共t兲 evolving with the fielddependent equations of motion 共12兲 is given by
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冉 冕
˜ 共⬁兲典 V 具J Q = T k BT 2
⬁
冊
N
0
共24兲 where subscript “0” emphasizes that ˜JQ共共s兲0兲 is obtained by solving the field-independent equations of motion.7,12 Two conditions are now imposed on the equations of motion 共12兲, and consequently on Ci and Di, for the determination of LQQ using Eq. 共9兲. These are 共i兲 preservation of zero total momentum,
冉 冕
˜ 共⬁兲典 具J V Q = T k BT 2
共25兲
and 共ii兲 equivalency of fluxes,
−
共26兲
J − B = JQ .
V k BT 2
⬁
˜ 共共s兲 兲 丢 ˜J 共0兲典 ds 具J Q 0 Q c
0
冕
⬁
冉 冕
⬁
冊
˜ 共共s兲 兲 丢 ˜J 共0兲典 ds F . 具J Q 0 Q c e
0
共27兲
As noted in Refs. 7 and 12, assuming that Fe is small, the ˜ 共⬁兲典 / T is directly proportional to the external quantity 具J Q field Fe with constant of proportionality equal to the phenomenological coefficient matrix LQQ / T2. It should be mentioned here that the condition given by Eq. 共26兲 can also be imposed by constructing Ci and Di so that B = 0 and J = JQ. As a consequence of this alternative construction, the equations of motion 共12兲 also satisfy adiabatic incompressibility of phase space AI⌫. IV. THERMAL CONDUCTIVITY ESTIMATION BY THE HNEMD METHOD
As mentioned in Sec. I, the HNEMD equations of motion have been employed in the estimation of GaN thermal conductivity in both bulk and nanowire systems.14,15 However, it is shown here that the original HNEMD method4,12 is not consistent for mixtures and alloys due to the nonequivalency of the fluxes J − B and JQ 共i.e., J − B ⫽ JQ兲. To appreciate this point, recall that Ci and Di in the original HNEMD equations of motion12 are given in the form N
Di = i −
Ci = 0,
1 兺 j . N j=1
1 J= V where
冉兺 冊 N
i=1
冉兺 N
pi N − 1 , mi N
pi Ti m i i=1
N
共32兲 or ˜ 共⬁兲典 1 具J Q = 2 共LQQ − LQA − LQB兲Fe , T T
共33兲
where LQA =
V kB
冕
V kB
冕
⬁
˜ 共共s兲 兲 丢 A ˜ 共0兲典 ds 具J Q 0 c
共34兲
˜ 共共s兲 兲 丢 B ˜ 共0兲典 ds. 具J Q 0 c
共35兲
0
and LQB =
⬁
0
Hence, estimation of LQQ requires the calculation of LQA + LQB in addition to running the standard HNEMD algorithm ˜ 共⬁兲典 / T. These extra correlation terms need to be to obtain 具J Q evaluated by means of a Green–Kubo method, which naturally involves the errors related to the integration and calculation of correlation function. If upon employing the Green– Kubo method LQA + LQB is found to be small compared to LQQ, then it is sufficient to use the classical HNEMD method given by Eq. 共28兲 for estimating LQQ.
共28兲
These choices are compatible with preservation of zero total momentum since 兺iDi = 0. On the other hand, in this case the fluxes B and J defined in Eqs. 共22兲 and 共23兲, respectively, are given by 1 B= V
冊
˜ 共共s兲 兲 丢 关A ˜ 共0兲 + B ˜ 共0兲兴典 ds F 具J Q 0 c e
0
Taking into account these two conditions, Eq. 共24兲 becomes ˜ 共⬁兲典 V 具J Q = T k BT 2
共31兲
is the difference between the heat flux JQ and the dissipative flux J. It can be seen from Eqs. 共29兲–共31兲 that the difference in fluxes J and B is not equal to the heat flux JQ, which violates the second condition 共26兲 in Sec. III. Therefore, in the HNEMD method, the long-time average of the macroscopic heat flux vector JQ in Eq. 共24兲 can be expressed with the aid of Eqs. 共29兲 and 共30兲 as
N
pi = 0, 兺 i=1
N
˜ 共t兲 1 兺 T 兺 pi A共⌫兲 = A VN j=1 j i=1 mi
˜ 共共s兲 兲 丢 关J ˜ 共0兲 − B ˜ 共0兲兴典 ds F , 具J Q 0 c e
共29兲 N
冊
1 pi − 兺 Tj 兺 = JQ − A, N j=1 i=1 mi
共30兲
V. M-HNEMD ALGORITHM
In this section, equations of motion are derived for a system of n different species which satisfy basic conditions 共25兲 and 共26兲 discussed in Sec. III. These equations are of form 共12兲, with particular choices for the tensor variables Ci and Di. In the following, a sequence of proposed forms for Ci and Di is suggested and refined to satisfy Eqs. 共25兲 and 共26兲, hence showing the development of the final expressions given at the end of this section. As observed in Sec. IV, using Di in form 共28兲 violates the equivalence of flux 共26兲. To circumvent this problem, Di may be initially modified as
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mi Di = i − 兺 j , ¯ j=1 m
共36兲
N ¯ = 兺i=1 mi. Using Eq. 共36兲 with Ci = 0, the flux vector where m J in Eq. 共23兲 becomes N
1 pi 1 − J = 兺 Ti V i=1 mi m ¯
冉兺 冊冉兺 冊 N
N
Tj
j=1
共37兲
pi
i=1
N pi = 0, provided that the total initial momentum is and 兺i=1 zero, hence J = JQ. However, the flux vector B in Eq. 共22兲 is now equal to
1 B= V
冉兺 N
i=1
N
冊
N
1 pi 1 pi − 兺 pi = ⫽ 0, 兺 mi m ¯ i=1 V i=1 mi
共38兲
resulting, again, in J − B ⫽ JQ. One way to achieve B = 0 is to choose Ci = − ri 丢
pi . mi
共39兲
However, this introduces additional terms in J, such that N
J = JQ + 兺 i=1
冉
冊
pi 丢 ri Fi ⫽ JQ . mi
N
共41兲
While Eqs. 共39兲 and 共41兲 enforce conditions 共25兲 and 共26兲, they do not satisfy frame invariance, i.e., they do not yield the same behavior under arbitrary translations and rotations of the system. To impose frame invariance, one should subtract from the position vector ri of every atom a suitably defined vector ¯r, which convects with the body. In this case, Eqs. 共39兲 and 共41兲 become
N
Di = i −
pi , mi
冉
N
共42兲
冊
mi m F j · 共r j − ¯r兲 I. 兺 j − Fi · 共ri − ¯r兲 − m¯i 兺 ¯ j=1 m j=1
There are various possible choices for ¯r which enforce frame invariance without violating Eqs. 共25兲 and 共26兲. One such N ¯ 兲兺i=1 miri, namely, the position vector choice is ¯r = rCM = 共1 / m of the mass center of the system. Other choices include the position vector of any fixed point in the system, provided that this point convects with any arbitrary translation and rotation of the whole system. One such choice would be, for example, the left-bottom corner ra of the system. However, all the aforementioned choices are incompatible with periodic boundary conditions 共PBCs兲. Indeed, in the case of PBCs, all atoms that exit the system from one side enter it again from the opposite side with the same momentum, see, e.g., Ref. 27 共Sec. 1.5.2兲. In this so-called wrapping process, terms 共42兲 with ¯r = rCM or ¯r = ra create large jumps in veloci-
pi , mi
冉
N
Di = i −
mi mi Di = i − 兺 j − 共Fi · ri兲I + 兺 共F j · r j兲I. ¯ ¯ j=1 m j=1 m
Ci = − 共ri − ¯r兲 丢
Ci = − 共ri − ¯ri兲 丢
共40兲
To simultaneously eliminate B and set J = JQ, one should also change Di to N
ties and forces, as ri −¯r is discontinuous in time as an atom re-enters the periodic cell with different values of Ci and Di than those with which it exited, see Sec. VI for numerical evidence of this problem. In this work, the effect of any discontinuities due to incompatibility of Ci and Di with PBCs is minimized by choosing for each atom an ¯r that depends only on atoms in its neighborhood. Specifically, the neighborhood of an atom is now comprised only of atoms within the cutoff radius rc ⬎ 0 of the potential. This treatment is still timediscontinuous due to neighboring atoms entering or exiting the cutoff region of a given atom. Clearly, the magnitude of this time-discontinuity depends crucially on the thermally driven motion in the material.28 For instance, in solid alloys transitions of atoms to/from the cutoff region can be quite rare and become virtually nonexistent for low enough temperatures. On the other hand, in dense fluids an atom may change neighbors much more frequently, see Sec. VI for further discussion. For this choice of ¯r, Eq. 共42兲 can be written as
N
共43兲
冊
mi mi j − Fi · 共ri − ¯ri兲 − 兺 F j · 共r j − ¯r j兲 I, 兺 ¯ j=1 ¯ j=1 m m N
i r j and Ni is the number of atoms in the where ¯ri = 共1 / Ni兲兺 j=1 j⫽i neighborhood of atom i within the cutoff radius. Despite the time-discontinuity, the choice of Eq. 共43兲 preserves total momentum and satisfies Eq. 共26兲 with B = 0 since the partial derivatives of Ci and Di with respect to the phase variables are unaffected. Therefore, the equations of motion 共12兲 with Eq. 共43兲 can be used to evaluate LQQ and are referred to as the M-HNEMD equations of motion under periodic boundary conditions. Also, Ci in Eq. 共43兲 remains bounded with increasing system size, unlike the choice of ¯r = rCM. Since B = 0, the equations of motion also satisfy the adiabatic incompressibility of phase space, as in the case of HNEMD method for single-species systems. A casual review reveals that when reduced to a singlespecies system 共mi = m兲, the M-HNEMD algorithm is not identical to the HNEMD algorithm. This demonstrates the nonuniqueness of the NEMD algorithms for evaluating thermal conductivity.
VI. RESULTS
The HNEMD and M-HNEMD algorithms are applied to estimate the thermal conductivity of Ar and GaN systems and the transport coefficient LQQ / T2 for an Ar–Kr system. A ˜ 共⬁兲典 corresponding sequence of the heat flux averages 具J Qx,k to a decreasing sequence of Fex,k is computed to determine the linear regime of the system under the action of the external field Fe = 共Fex,k , 0 , 0兲. In this case, xx is estimated for Ar and GaN and LQQ,xx / T2 is estimated for Ar–Kr systems using the slope method described in Ref. 7. Before the external field Fe is switched on, the system is equilibrated using the NH thermostat which generates the 共extended兲 canonical en-
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0.14
κ [W/m−K]
0.12 0.1
HNEMD: κxx HNEMD: κyx HNEMD: κzx M−HNEMD: κxx M−HNEMD: κyx M−HNEMD: κzx
0.08 0.06 0.04 0.02 0 −0.02
0
0.02 0.04 0.06 0.08 0.1 Fex [1/Å]
NORMALIZED CORRELATION
0.16
0.12 0.14 0.16
0.8 0.6 0.4 0.2 0 −0.2
0
2
4
(a) Variation of thermal conductivities κxx , κyx and κzx
HNEMD: JQx HNEMD: JQy HNEMD: JQz M−HNEMD: JQx M−HNEMD: JQy M−HNEMD: JQz
0.015
0.12
0.01
0.1
0.005 0 −0.005
10
0.14
κ [W/m−K]
〈JQ〉/T [W/m−K−Å]
0.02
8
(a) Normalized Ar auto-correlation function J˜Qx (0)J˜Qx (t) . J˜Qx (0)J˜Qx (0)
with Fex . 0.025
6 t [ps]
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Fex [1/Å]
˜ Q (∞) J (b) Variation of the components of with Fex . T FIG. 1. Comparison of the M-HNEMD and HNEMD algorithms for Ar at the triple point. The dashed and solid lines correspond respectively to the M-HNEMD and HNEMD estimates of xx.
0.08 0.06 0.04 0.02 0
0
2
4
6
8
10
t [ps]
(b) Integral of the auto-correlation function leading to thermal conductivity estimate κxx .
semble given in Eq. 共13兲. When the external field is applied, the duration of each of the simulations in the sequence is ˜ 共⬁兲典 negchosen to make the variance of the estimated 具J Qx,k ligible and hence the reported error estimates are based on the regression errors. The time integration scheme used for the HNEMD method is described in Ref. 7 and is modified for the M-HNEMD method, as in Appendix. A. Argon
As an initial test, the M-HNEMD algorithm with Eq. 共43兲 is applied to a single-species 共n = 1兲 Ar system at its triple point 共T = 86.5 K, face-centered cubic lattice of length 5.719 Å兲. The system is modeled using a Lennard-Jones potential 关M = 2 in Eq. 共15兲兴 with N = 256 atoms, cutoff radius of rc = 13 Å, and periodic boundary conditions on the 4 ⫻ 4 ⫻ 4 unit cells. In the potential, the characteristic energy is ⑀ = 1.6539⫻ 10−21 J and the characteristic length is = 3.405 Å.29 Each run is carried out for 106 time steps with a step-size of ⌬t = 4.0 fs. The computationally tractable linear regime of Fex is found to be between 0.001 Å−1 ⬍ Fex ⬍ 0.1 Å−1, as shown in ˜ 共⬁兲典 / TF 兴 is independent of Fig. 1共a兲, where xx,k = 关具J Qx,k ex,k Fex,k. It is concluded from Fig. 1共a兲 that the results from the M-HNEMD and HNEMD algorithms are practically identical, thus validating the M-HNEMD algorithm for a singlespecies system. The value of thermal conductivity using the
FIG. 2. Green–Kubo estimate of the thermal conductivity of Ar based on the direct integration of the heat flux autocorrelation function.
M-HNEMD algorithm with the slope method is found to be 0.1305⫾ 0.0002 W / mK, see Fig. 1共b兲. This compares favorably to the HNEMD result 0.1302⫾ 0.0002 W / mK and the Green–Kubo result 0.129 W/mK, see Fig. 2. As an aside, it is also observed that when using ¯r = rCM in Eq. 共42兲 the thermal conductivity xx shows a decreasing trend for increasing values of Fex until it reaches a regime of severe discontinuity, see Fig. 3. As argued in Sec. V, the discontinuity of xx共Fex兲 may be attributed to the nonsmooth transition during the remapping due to the PBCs. This problem is eliminated with the use of Eq. 共43兲. Interestingly, the use of Eq. 共43兲 also eliminates the decreasing trend of xx, which facilitates the identification of a linear regime.
B. Argon-krypton
The M-HNEMD algorithm 共43兲 is applied to the binary Ar–Kr mixture system at T = 115.6 K. The system is comprised of a total of N = 256 atoms 共128 Ar and 128 Kr兲 in a cube of volume 24.1923 Å3 with PBCs. It is modeled using a Lennard-Jones potential, where the parameters for Kr are ⑀ = 2.3056⫻ 10−21 J and = 3.670 Å.30 The two parameters
1
κxx κyx κzx
1.5
0.2 0.15 0.1 0.05 0
1 0.5
0
NORMALIZED CORRELATION
2
0.04 0.08
0 −0.5 −1 0
0.02
0.04
0.06 0.08 Fex [1/Å]
0.1
LQQ/T2 [W/m−K]
for Ar–Kr pairs are determined from the Lorentz–Berthelot mixing rules.31 The equations of motion are integrated with step-size of ⌬t = 4.0 fs for 106 time steps. Figure 4共a兲 shows that the linear regime is found approximately when 0.01 Å−1 ⬍ Fex ⬍ 0.085 Å−1 for both the M-HNEMD and HNEMD algorithms. The M-HNEMD estimate for the transport coefficient LQQ,xx / T2 is found to be 0.087 43⫾ 0.000 11 W / mK, see Fig. 4共b兲. This is in good agreement with the Green–Kubo estimate of 0.087 20 W/mK at a correlation time of t = 1 ps, see Fig. 5. The HNEMD
〈JQx(0).JQx(t)〉 〈JQx(0).Ax(t)〉
0.4
〈JQx(0).Bx(t)〉
0.2 0 0
2
4
6
8
10
(a) Correlations normalized by J˜Qx (0)J˜Qx (0). 0.1 0.09 0.08 0.07 2
0.06
LQQ,xx/T
0.05
LQA,xx/T2
0.04
LQB,xx/T2
0.03 0.02 0.01 0 −0.01
0
2
4
6
8
10
t [ps]
(b) Integrals of the correlation functions. HNEMD: LQQ,xx/T2 HNEMD: LQQ,yx/T2 HNEMD: LQQ,zx/T2 M−HNEMD: LQQ,xx/T2 M−HNEMD: LQQ,yx/T2 2 M−HNEMD: LQQ,zx/T
0
0.02 0.04 0.06 0.08 0.1 Fex [1/Å]
0.12 0.14 0.16
(a) Variation of the transport coefficients LQQ,xx , LQQ,yx and LQQ,zx with Fex . 0.016
HNEMD: JQx HNEMD: JQy HNEMD: JQz M−HNEMD: JQx M−HNEMD: JQy M−HNEMD: JQz
0.014 〈JQ〉/T [W/m−K−Å]
0.6
t [ps]
FIG. 3. Variation of thermal conductivities xx, yx, and zx with Fex for Ar at the triple point for ¯r = rCM, demonstrating an instability at approximately Fex = 0.089 Å−1 and a significant decreasing trend 共see inset兲.
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01
0.8
−0.2
0.12
0.012 0.01 0.008
˜ 共0兲J ˜ 共t兲典, 具J ˜ 共0兲A ˜ 共t兲典, and FIG. 5. Ar–Kr correlation functions 具J Qx Qx Qx x ˜ 共0兲B ˜ 共t兲典, as well as integration of the correlation functions leading to 具J Qx x LQQ,xx / T2, LQA,xx / T2, and LQB,xx / T2 estimates using the Green–Kubo method.
estimate is found to be 0.086 18⫾ 0.000 13 W / mK, which is very close to the Green–Kubo and M-HNEMD estimates, see Fig. 5. As mentioned in Sec. IV, since 共LQA,xx + LQB,xx兲 / T2 = 0.000 43 W / mK is small compared to the Green–Kubo estimate from Fig. 5, either the HNEMD or the M-HNEMD algorithm may be employed to obtain an accurate estimate of LQQ,xx for Ar–Kr in the given state. Figure 6 shows the magnitude of the discontinuities in 15
0.006
14
0.004 0.002 0 −0.002
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Fex [1/Å]
(b) Variation of the components of the heat flux vector ˜ Q (∞) J with Fex . T FIG. 4. Comparison of the M-HNEMD and HNEMD algorithms for Ar–Kr. The dashed and solid lines correspond to the M-HNEMD method and HNEMD estimates of LQQ,xx / T2, respectively.
X POSITION [A]
κ [W/m−K]
J. Chem. Phys. 133, 034122 共2010兲
HNEMD for mixtures and alloys
CORRELATION INTEGRAL [W/m−K]
034122-7
13 12 11 10
− ri ri
9 0
20
40
60
80 100 120 140 160 180 200 TIME [ps]
FIG. 6. Trajectories ri and ¯ri of a typical atom i in an Ar–Kr system.
800
0.007
2
HNEMD: LQQ,xx/T 2 HNEMD: LQQ,yx/T HNEMD: LQQ,zx/T2 2 M−HNEMD: LQQ,xx/T 2 M−HNEMD: LQQ,yx/T 2 M−HNEMD: LQQ,zx/T
700 600 κ [W/m−K]
J. Chem. Phys. 133, 034122 共2010兲
Mandadapu, Jones, and Papadopoulos
500 400
0.006 0.005
〈 JQ 〉 [eV/Å2−ps]
034122-8
300 200 100 0 −100
0.004 0.003
〈JQx〉 〈JQy〉 〈JQz〉
0.002 0.001 0 −0.001
0
5e−05
0.0001 Fex [1/Å]
0.00015
0.0002
−0.002 −0.003 0
(a) Variation of the thermal conductivities κxx , κyx and κzx with Fex . 0.05
〈JQ〉/T [W/m−K−Å]
0.03 0.02 0.01 0 −0.01
0
5e−05
0.0001 Fex [1/Å]
1
1.5
2
2.5
3
3.5
4
t [ns]
FIG. 8. The running average of the components of the heat flux vector JQ for GaN system for Fex = 1.3⫻ 10−4 Å−1 using the M-HNEMD method.
HNEMD: JQx HNEMD: JQy HNEMD: JQz M−HNEMD: JQx M−HNEMD: JQy M−HNEMD: JQz
0.04
0.5
0.00015
0.0002
(b) Variation of the components of the heat flux vector ˜ Q (∞) J with Fex . T FIG. 7. Comparison of the M-HNEMD and HNEMD algorithms for GaN. The dashed and solid lines correspond to the M-HNEMD method and HNEMD estimates of xx, respectively.
the trajectory of a typical ¯ri relative to the trajectory of the associated atom ri. These trajectories were captured before the field Fex was applied to allow for comparison of ¯ri to an unperturbed ri. Even in a dense fluid, it is apparent that the discontinuities are quite small relative to the normal fluctuations in position for the chosen time-step.
C. Gallium-nitride
The HNEMD and M-HNEMD methods are employed to estimate the thermal conductivity of a perfect GaN system 共wurtzite crystal structure兲 using an orthogonal cell with lattice parameters a = 3.19 Å and c = 5.20 Å at T = 500 K—see Ref. 6 for details on the lattice structure. The system is modeled using 512 atoms with four unit cells per direction 共4 ⫻ 4 ⫻ 4兲 under periodic boundary conditions and a Stillinger–Weber potential 关M = 3 in Eq. 共15兲兴—see Refs. 32 and 33. It is observed that 4 ⫻ 4 ⫻ 4 system is sufficient for obtaining the value of thermal conductivity using both HNEMD and M-HNEMD algorithms. A time-step of 0.5 fs is used for 8 ⫻ 106 time steps. The linear regime in the HNEMD method is found in the range 7 ⫻ 10−5 Å−1 ⬍ Fez ⬍ 1.3⫻ 10−4 Å−1 and the thermal conductivity is estimated via the slope method as 133.65⫾ 2.50 W / mK, see Figs. 7共a兲 and 7共b兲.
Using the M-HNEMD method, the thermal conductivity is found to be 132.02⫾ 5.85 W / mK, see Figs. 7共a兲 and 7共b兲, which is comparable to the HNEMD result. Both the HNEMD and M-HNEMD estimates are comparable to the estimate of the direct method estimate, which is 120.59 W/mK.6 It can also be seen in Fig. 8 that the average ˜ 共t兲典 converges after 4 ⫻ 106 time steps for of the heat flux 具J Q −4 Fex = 1.0⫻ 10 Å−1 thus demonstrating that the duration of the simulations is sufficient. Finally, the Green–Kubo approach yields an estimate of 128.09 W/mK using 8 ⫻ 106 time steps and 1000 atoms 共5 ⫻ 5 ⫻ 5兲. In the Green–Kubo approach, it is observed that a system at least as large as 5 ⫻ 5 ⫻ 5 is needed to obtain a limiting thermal conductivity value. In all cases, the Green–Kubo estimate is based on the criterion of the first plateau attained by the integral of the ˜ 共t兲J ˜ 共0兲典 correlation, as in Ref. 3. Figure 9 shows that 具J Q Q decays in a complicated manner with correlation time, unlike the Ar–Kr system, which makes this criterion somewhat ambiguous. As seen in Fig. 9, the noise past a correlation time of approximately 200 ps is quite significant, thus requiring considerably more simulation time than the M-HNEMD method to obtain a similar quality thermal conductivity estimate. However, 共LQA,xx + LQB,xx兲 / T2 = 0.051 W / mK is extremely small compared to the Green–Kubo estimate leading to the conclusion that the HNEMD and M-HNEMD methods yield comparable results for GaN. VII. DISCUSSION
As observed in Sec. VI, the Green–Kubo method may not be viable for the estimation of thermal conductivity in systems such as GaN. In such systems, it is difficult to establish that the complex autocorrelation function has decayed sufficiently to permit accurate integration. In addition, given that the decay of the autocorrelation function is fairly long 共at least 150–200 ps in the case of GaN兲, it takes considerable simulation time 共of the order of 8 ns in GaN兲 to obtain the thermal conductivity within reasonable statistical certainty. These problems may be avoided by using the HNEMD and M-HNEMD methods, which involve the average of the heat flux vector alone as opposed to the integration of the noisy autocorrelation of the heat flux vector. Also, these nonequilibrium methods yield results of reasonable ac-
034122-9 1 NORMALIZED CORRELATION
J. Chem. Phys. 133, 034122 共2010兲
HNEMD for mixtures and alloys
〈JQx(0).JQx(t)〉
0.8 0.6
〈JQx(0).Ax(t)〉
0.4
〈JQx(0).Bx(t)〉
0.2 0
1
−0.2 −0.4
0
−0.6 −0.8
−1 0
50
100
150
200
250
300
t [ps]
CORRELATION INTEGRAL [W/m−K]
(a) Correlations normalized by J˜Qx (0)J˜Qx (0). 140 120 100 80
LQQ,xx/T2
60
LQA,xx/T2
40
LQB,xx/T2
20 0 −20
0
50
100
150
200
250
t [ps]
(b) Integrals of the correlation functions ˜ 共0兲J ˜ 共t兲典, 具J ˜ 共0兲A ˜ 共t兲典, and FIG. 9. GaN correlation functions 具J Qx Qx Qx x ˜ 共0兲B ˜ 共t兲典, as well as integration of the correlations functions leading to 具J Qx x xx = LQQ,xx / T2, LQA,xx / T2, and LQB,xx / T2 estimates using the Green–Kubo method. Note that in 共a兲 the data have been decimated to show each correlation clearly; the inset shows the full correlations which densely fill the envelopes described by the decimated data.
curacy for relatively small total simulation times, as the signal-to-noise ratio is high due to the action of the finite nonzero external field Fe. For single-component systems, the HNEMD and M-HNEMD algorithms, while not mathematically identical, yield statistically identical results, which demonstrates the nonuniqueness in the construction of nonequilibrium molecular dynamics methods. Interestingly, even when applied to multicomponent Ar–Kr and GaN systems, the HNEMD and M-HNEMD algorithms yield very similar results, as LQA + LQB is very small compared to the heat transport coefficient tensor LQQ. However, the M-HNEMD is theoretically consistent with the linear response theory for multicomponent system, in the sense that in addition to satisfying preservation of total momentum 关see Eq. 共25兲兴 and being compatible with periodic boundary conditions, it also satisfies equivalency of fluxes 关see Eq. 共26兲兴. In contrast, for multicomponent systems the HNEMD method violates Eq. 共26兲, thereby requiring additional equilibrium simulations to establish that LQA + LQB is small compared to LQQ. Although these results have not answered the question whether these quantities are insignificant for all material systems under all
conditions conclusively, it does suggest that the original HNEMD method can provide accurate thermal conductivity estimates without strictly satisfying the equivalence of fluxes and the incompressibility of phase space. Nevertheless, in all cases, without resorting to additional simulations to calculate expensive time-correlations, the marginally more complex M-HNEMD algorithm will provide accurate results if used in the linear regime. As expected, the M-HNEMD method shares some of the same challenges that apply to other nonequilibrium methods that rely on the linear response theory. First, it requires simulations corresponding to a decreasing sequence of Fe to establish the linear regime. This can be alleviated by performing these simulations in parallel, as they are completely independent of each other. Second, it becomes extremely inefficient for very small Fe, where the signal-to-noise ratio is low. In such a case, it takes simulation time on the order of a Green–Kubo estimate to obtain a reasonable estimate of ˜ 共⬁兲典 / TF as F tends to zero. Other NEMD methods 具J Qx ex ex may become applicable in this case, such as the one by Ciccotti et al.,34,35 which employed a direct difference of heat flux between unperturbed and periodically perturbed trajectories to obtain the linear response of the system. In the NEMD under consideration, it is important to set Fex far enough from zero to avoid this problem, while still staying within the linear regime. An estimate of the linear regime can be obtained from prior knowledge of the phonon mean-free path.7 Third, and most important, the width of the linear regime decreases with an increasing size of the system, thereby making it difficult to obtain a reliable thermal conductivity estimate.36 This problem may be rectified by adopting the methodology in Ref. 36, which has been shown to maintain a constant width of the linear regime. ACKNOWLEDGMENTS
This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Co., for the United States Department of Energy under Contract No. DE-ACO4-94AL85000. APPENDIX: NUMERICAL INTEGRATION ALGORITHM
Following the work in Ref. 7 the governing system of Eq. 共12兲 can be decomposed into
冤 冥 冤 共 兲冥 冤 冥 冤
ri d pi = dt
0 0
1 ¯T共p兲 Q T
−1
0 0 + − pi + Di共r,p兲Fe 0 0 L2
L1
冤 冥冤 冥冤
0 + Fi共r兲 + 0
1 m pi
L4
L5
0 0
+
Ci共r,p兲Fe 0 0 L6
冥
L3
冥
,
共A1兲
034122-10
J. Chem. Phys. 133, 034122 共2010兲
Mandadapu, Jones, and Papadopoulos
N where ¯T共p兲 = 共1 / 3NkB兲兺i=1 共pi · pi / mi兲. The operator-split method in Ref. 37 is employed to integrate the preceding system of ordinary differential equations. In this method, the propagation
⌫共t兲 = exp共iLt兲⌫共0兲
共A2兲
of the initial state is approximated according to
冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊
exp共iL⌬t兲 = exp iL1
⌬t ⌬t ⌬t exp iL2 exp iL3+4 2 2 2
⫻exp iL5
⌬t ⌬t exp共iL6⌬t兲exp iL5 2 2
⫻exp iL3+4
⌬t ⌬t ⌬t exp iL2 exp iL1 , 2 2 2 共A3兲
where higher-order terms have been omitted and iL is the p-Liouvillean 共Ref. 5, Sec. 3.3兲. The update formulas for the individual steps are defined as L1:n+1/2 = n +
冉
冉
˜ n+1/2 = exp − L2:p
共A4兲
⌬t n+1/2 pn , 2
冊
共A5兲
⌬t Fn , 2m
共A6兲
L3+4:pn+1/2 = ˜pn+1/2 +
L5:rn+1/2 = rn +
冊
⌬t 1 ¯T共pn兲 −1 , 2 Q T
⌬t pn+1/2 , 2m
共A7兲
˜ n+1 = rn+1/2 + ⌬tC共rn+1/2,pn+1/2兲Fe , L6:r L5:rn+1 = ˜rn+1 +
⌬t pn+1/2 , 2m
共A8兲 共A9兲
˜F = F共r 兲, n+1 n+1
共A10兲
˜ 兲F , Fn+1 = ˜Fn+1 + D共rn+1,pn+1/2,F n+1 e
共A11兲
˜ n+1 = pn+1/2 + L3+4:p
冉
⌬t Fn+1 , 2m
共A12兲
冊
L2:pn+1 = exp −
⌬t n+1/2 ˜pn+1 , 2
L1:n+1 = n+1/2 +
⌬t 1 ¯T共pn+1兲 −1 , 2 Q T
冉
共A13兲
冊
共A14兲
where, in the interest of brevity, all the i subscripts referring to atoms have been omitted. Here the subscript n refers to the time-step. Also note that F0 is defined as F0 = F共r0兲, given initial conditions r0 and p0.
All the steps of this algorithm involve exact integration of the respective propagators, with the exception of Eqs. 共A8兲 and 共A12兲 which involve integration of the forces CiFe and DiFe. Indeed, Eq. 共A12兲 is exact except for the part of Di that depends on kinetic energy and, consequently, on terms involving squares of the components of pi. Moreover, Di involves the momenta of each atom through the average kinetic energy embedded in Di. Likewise, Eq. 共A8兲 is inexact due to the dependence of Ci on a set of r j, which couples the integration for atom i to the motions of its neighboring atoms. W. G. Hoover, Computational Statistical Mechanics 共Elsevier, Amsterdam, 1991兲. 2 S. G. Volz and G. Chen, Phys. Rev. B 61, 2651 共2000兲. 3 P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B 65, 144306 共2002兲. 4 D. J. Evans, Phys. Lett. A 91, 457 共1982兲. 5 D. J. Evans and G. P. Morris, Statistical Mechanics of Non-equilibrium Liquids 共Academic, New York, 1990兲. 6 X. W. Zhou, S. Aubry, R. E. Jones, A. Greenstein, and P. K. Schelling, Phys. Rev. B 79, 115201 共2009兲. 7 K. K. Mandadapu, R. E. Jones, and P. Papadopoulos, J. Chem. Phys. 130, 204106 共2009兲. 8 J. M. Luttinger, Phys. Rev. 135, A1505 共1964兲. 9 M. J. Gillan and M. Dixon, J. Phys. C 16, 869 共1983兲. 10 P. J. Daivis and D. J. Evans, Mol. Phys. 81, 1289 共1994兲. 11 A. Perronace, J.-M. Simon, B. Rousseau, and G. Ciccotti, Mol. Phys. 99, 1139 共2001兲. 12 K. K. Mandadapu, R. E. Jones, and P. Papadopoulos, Phys. Rev. E 80, 047702 共2009兲. 13 P. J. Daivis and D. J. Evans, Chem. Phys. 198, 25 共1995兲. 14 Z. Wang, X. Zu, F. Gao, W. J. Weber, and J.-P. Crocombette, Appl. Phys. Lett. 90, 161923 共2007兲. 15 Z. Wang, F. Gao, J. P. Crocombette, X. T. Zu, L. Yang, and W. J. Weber, Phys. Rev. B 75, 153303 共2007兲. 16 D. J. Evans and P. T. Cummings, Mol. Phys. 72, 893 共1991兲. 17 S. R. de Groot, Thermodynamics of Irreversible Processes 共Interscience, New York, 1951兲. 18 L. Onsager, Phys. Rev. 37, 405 共1931兲. 19 L. Onsager, Phys. Rev. 38, 2265 共1931兲. 20 R. Zwanzig, J. Chem. Phys. 40, 2527 共1964兲. 21 D. J. Evans and B. L. Holian, J. Chem. Phys. 83, 4069 共1985兲. 22 W. G. Hoover, Phys. Rev. A 31, 1695 共1985兲. 23 It should be noted that the property of the Nosé–Hoover thermostat to sample the canonical phase space distribution relies on the system being ergodic under the Nosé–Hoover dynamics. For systems that are not ergodic in this manner, e.g., certain small or stiff systems, the technique of Nosé–Hoover chains 共Ref. 38兲 provides a remedy. It is apparent that the systems analyzed in this work are well-behaved in this respect and hence Nosé–Hoover chains are not applied. 24 J. W. Martin, J. Phys. C 8, 2837 共1975兲. 25 J. H. Irving and J. G. Kirkwood, J. Chem. Phys. 18, 817 共1950兲. 26 R. J. Hardy, J. Chem. Phys. 76, 622 共1982兲. 27 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids 共Oxford University Press, New York, 1987兲. 28 The jumps in ¯ri are proportional to 共n / Ni兲rc, where n is the net efflux of atoms from the cutoff region over a time-step. Given a well-defined density, the long-time expectation of n / Ni is zero; but on shorter time-scales, n will be proportional to the surface area of the cutoff region and be affected by the diffusion and density fluctuation characteristics of the material at the given thermodynamic state. Specifically, n will be related to the spatiotemporal two-point density correlation function of the system integrated over the cutoff region. 29 L. A. Rowley, D. Nicholson, and N. G. Parsonage, J. Chem. Phys. 17, 401 共1975兲. 30 M. Moosavi and E. K. Gaharshadi, Fluid Phase Equilib. 274, 51 共2008兲. 31 R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4th ed. 共McGraw-Hill, New York, 1987兲. 32 A. Béré and A. Serra, Phys. Rev. B 65, 205323 共2002兲. 1
034122-11
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